Question 187798
We basically have this triangle set up:



{{{drawing(500,500,-0.5,2,-0.5,3.2,
line(0,0,0,3),
line(0,3,2,0),
line(2,0,0,0),
locate(-0.2,1.5,x+1),
locate(1,-0.2,x+1),
locate(1,2,2x-1)
)}}}


Remember, the Pythagorean Theorem is {{{a^2+b^2=c^2}}} where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since both legs are {{{x+1}}} this means that {{{a=x+1}}} and {{{b=x+1}}}


   

Also, since the hypotenuse is {{{2x-1}}}, this means that {{{c=2x-1}}}.




{{{a^2+b^2=c^2}}} Start with the Pythagorean theorem.



{{{(x+1)^2+(x+1)^2=(2x-1)^2}}} Plug in {{{a=x+1}}}, {{{b=x+1}}}, and {{{c=2x-1}}} 



{{{x^2+2x+1+x^2+2x+1=4x^2-4x+1}}} FOIL



{{{2x^2+4x+2=4x^2-4x+1}}} Combine like terms.



{{{2x^2+4x+2-4x^2+4x-1=0}}} Get all terms to the left side.



{{{-2x^2+8x+1=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=-2}}}, {{{b=8}}}, and {{{c=1}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(8) +- sqrt( (8)^2-4(-2)(1) ))/(2(-2))}}} Plug in  {{{a=-2}}}, {{{b=8}}}, and {{{c=1}}}



{{{x = (-8 +- sqrt( 64-4(-2)(1) ))/(2(-2))}}} Square {{{8}}} to get {{{64}}}. 



{{{x = (-8 +- sqrt( 64--8 ))/(2(-2))}}} Multiply {{{4(-2)(1)}}} to get {{{-8}}}



{{{x = (-8 +- sqrt( 64+8 ))/(2(-2))}}} Rewrite {{{sqrt(64--8)}}} as {{{sqrt(64+8)}}}



{{{x = (-8 +- sqrt( 72 ))/(2(-2))}}} Add {{{64}}} to {{{8}}} to get {{{72}}}



{{{x = (-8 +- sqrt( 72 ))/(-4)}}} Multiply {{{2}}} and {{{-2}}} to get {{{-4}}}. 



{{{x = (-8 +- 6*sqrt(2))/(-4)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (-8+6*sqrt(2))/(-4)}}} or {{{x = (-8-6*sqrt(2))/(-4)}}} Break up the expression.  



{{{x = (4+3*sqrt(2))/(2)}}} or {{{x = (4+3*sqrt(2))/(2)}}} Reduce



So the answers are {{{x = (4+3*sqrt(2))/(2)}}} or {{{x = (4+3*sqrt(2))/(2)}}} 



which approximate to {{{x=-0.121}}} or {{{x=4.121}}} 



Since {{{x=-0.121}}} generates a negative length, we'll ignore this answer.



So the only solution is {{{x=4.121}}}



This means that the two legs are


{{{x+1=4.121+1=5.121}}}



And the hypotenuse is 


{{{2(4.121)-1=8.242-1=7.242}}}


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Answer:



So the legs are approximately 5.121 units long and the hypotenuse is about 7.242 units long